Reciprocal Fibonacci constant

The reciprocal Fibonacci constant ψ is the sum of the reciprocals of the Fibonacci numbers:

$$\psi =\sum _{k=1}^{\mathrm{\infty }}\frac{1}{F_k}=\frac{1}{1}+\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+\frac{1}{8}+\frac{1}{13}+\frac{1}{21}+\cdots .$$

Because the ratio of successive terms tends to the reciprocal of the golden ratio, which is less than 1, the ratio test shows that the sum converges.

The value of ψ is approximately

$$\psi =3.359885666243177553172011302918927179688905133732\dots$$ (sequence A079586 in the OEIS).

With k terms, the series gives O(k) digits of accuracy. Bill Gosper derived an accelerated series which provides O(k²) digits. ψ is irrational, as was conjectured by Paul Erdős, Ronald Graham, and Leonard Carlitz, and proved in 1989 by Richard André-Jeannin.

Its simple continued fraction representation is:

$$\psi =[3;2,1,3,1,1,13,2,3,3,2,1,1,6,3,2,4,362,2,4,8,6,30,50,1,6,3,3,2,7,2,3,1,3,2,\dots ]$$ (sequence A079587 in the OEIS).

Generalization and related constants

In analogy to the Riemann zeta function, define the Fibonacci zeta function as $${\zeta }_F(s)=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{(F_n{)}^s}=\frac{1}{1^s}+\frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{5^s}+\frac{1}{8^s}+\cdots$$ for complex number s with Re(s) > 0, and its analytic continuation elsewhere. Particularly the given function equals ψ when s = 1.

It was shown that:

• The value of ζ_F(2s) is transcendental for any positive integer s, which is similar to the case of even-index Riemann zeta-constants ζ(2s).
• The constants ζ_F(2), ζ_F(4) and ζ_F(6) are algebraically independent.
• Except for ζ_F(1) which was proved to be irrational, the number-theoretic properties of ζ_F(2s + 1) (whenever s is a non-negative integer) are mostly unknown.

See also

• List of sums of reciprocals
• List of mathematical constants